On Induced Colourful Paths in Triangle-free Graphs

TL;DR

This paper investigates the existence of induced colourful paths in properly coloured triangle-free graphs, proving the conjecture for graphs with girth at least their chromatic number and discussing related recent results.

Contribution

It proves the conjecture that such graphs contain induced colourful paths on their chromatic number when girth is sufficiently large, extending previous work.

Findings

Confirmed the conjecture for graphs with girth at least (G)

Established conditions under which induced colourful paths exist

Connected results to recent research on induced paths in triangle-free graphs

Abstract

Given a graph $G=(V,E)$ whose vertices have been properly coloured, we say that a path in $G$ is "colourful" if no two vertices in the path have the same colour. It is a corollary of the Gallai-Roy-Vitaver Theorem that every properly coloured graph contains a colourful path on $\chi(G)$ vertices. We explore a conjecture that states that every properly coloured triangle-free graph $G$ contains an induced colourful path on $\chi(G)$ vertices and prove its correctness when the girth of $G$ is at least $\chi(G)$. Recent work on this conjecture by Gy\'arf\'as and S\'ark\"ozy, and Scott and Seymour has shown the existence of a function $f$ such that if $\chi(G)\geq f(k)$, then an induced colourful path on $k$ vertices is guaranteed to exist in any properly coloured triangle-free graph $G$.